136 KANSAS UNIVERSITY QUARTERLY. 
ee Von de a TEN he ed Ry Cy are ea 
a St S@e 4 An pea ace sere malt a Mb Ve por 
O) Gye spc) Ones pi em L/P Gee) 
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= +ka-— a* + ha; 
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al mea, ah oT Ee ON crate | Ve etl Ode: Fcc 
TADS COl stile Rap nec eO ipa © Sig. 8 ee) 
GoM Og Oly tle sult MOmMOl ea A Ol Oe sul 
I I 
= Ve 
Ix, V4 Z, 1) XxX y Z I 
Ae ME Mega Neeley we at 
Tap AG) okO) AT a eG ae. 
17 ore f [POOR aer 
Ix valez 1| satan pte wAry os 
Hetil rarer Rees (Aieealy sean ea 
ee Qe 0 [. ap ee AO 
OO aaa ofkm , OOF Bl ies 5) meee 
ma = - = a’ na) = | seas eee a*-+9a, 
cAyaiynasas oad € Xe AV EZ 2 Pal # 
Ee ar Crema ae JDC iu 
Is 79" Ce tO Ios Pence] 
te eC FOr eel Ly fOr Oat 
The coordinates of the invariant point A are (a,b,c); the direc- 
tion of the line Al is determined by p and q; f determines the 
position of the invariant plane through the line Al. k, h, anda 
have the same meanings as in type III of the plane. m,nand g 
have analogous meanings with respect to the bundle of invariant 
twisted cubics which constitute the path curves of every one para- 
meter group of transformations of this type. 
TYPE VI. 
Sy Seat iy MN Zed ea 
Bie oy. Se. at eee lby ey si 
oi O.. © San OY © 
I O 
X,—a x—a y,—b y—b Cee a Oe ae 
— = —— Kk : 
B= 6) G2==C G56 “20 5 Spage Ly Sti 25 | 
1 1 a ae ee Ke ty see or 
events Petco rat ea Ove One 
[Ce A eso 5:0 ii 1m © © 
Koya tay tu (6) Onmeul opie 
The coordinates of the invariant point D are (a,b,c); 1, m, n 
fix the position of the plane of invariant points. k is the anhar- 
monic ratio along each invariant line through the point D. 
